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In rational functions like \(s(x)=\frac{2x+3}{x-1}\), horizontal asymptotes help us understand how the function behaves as \(x\) moves towards infinity or negative infinity. To find the horizontal asymptote, we look at the degrees of the numerator and denominator. These are the highest powers of \(x\) in each polynomial.
If the degrees are equal, as with our function where both are 1 (\(n=m=1\)), the horizontal asymptote can be found by dividing the leading coefficients. Here, it's \(y = \frac{2}{1} = 2\). The horizontal asymptote tells us that as \(x\) becomes very large (positively or negatively), \(s(x)\) will approach 2.
This means that the graph of the function will get closer and closer to the line \(y=2\) but never actually touch it.

Vertical asymptotes are different from horizontal ones. They tell us where the function goes to infinity or negative infinity. In simpler terms, these occur where the function is undefined due to division by zero.
For the function \(s(x)=\frac{2x+3}{x-1}\), vertical asymptotes occur where the denominator becomes zero, making the function undefined at those points.

• We set the denominator \(x-1\) of \(s(x)\) equal to 0: \(x-1=0\).
• Solving this gives \(x=1\).

This means that there is a vertical asymptote at \(x=1\). At this \(x\)-value, the function \(s(x)\) will increase or decrease without bound. On the graph, you'll see a vertical line at \(x=1\) where the function appears to shoot up to infinity.

Rational functions are expressions formed by dividing two polynomials. They often have interesting properties, like asymptotes. In our example, \(s(x)=\frac{2x+3}{x-1}\), the rational function is the ratio of a linear polynomial in the numerator and another in the denominator.
Rational functions can have:

• Horizontal asymptotes, which reflect the end behavior of the function as \(x\) goes to infinity.
• Vertical asymptotes, indicating where the function is undefined due to division by zero.

Understanding rational functions involves determining these asymptotes, as they play a key role in sketching their graphs. Horizontal and vertical asymptotes guide us in predicting the behavior of rational functions, showing how they behave at the extremes of their domain. Mastering these concepts is crucial for more advanced math topics.